A206291 -id:A206291 - cp's OEIS frontend

A004615 Divisible only by primes congruent to 1 mod 5.

1, 11, 31, 41, 61, 71, 101, 121, 131, 151, 181, 191, 211, 241, 251, 271, 281, 311, 331, 341, 401, 421, 431, 451, 461, 491, 521, 541, 571, 601, 631, 641, 661, 671, 691, 701, 751, 761, 781, 811, 821, 881, 911, 941

Offset: 1

N. J. A. Sloane

nonn

Also numbers with all divisors ending with digit 1.
Union of number 1, A030430 and A068872. - Jaroslav Krizek, Feb 12 2012
Also numbers with all divisors ending with the same digit; as 1 divides all the integers, this digit is necessarily 1 (see first comment); hence, for these numbers m: A330348(m) = A000005(m). - Bernard Schott, Nov 09 2020

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A027748, A030430 (primes), A068872 (composites).

Cf. A010879, A027750, A002808, A330348, A338784 (subsequence).

a004615 n = a004615_list !! (n-1)
a004615_list = filter (all (== 1) . (map (`mod` 5) . a027748_row)) [1..]
-- Reinhard Zumkeller, Apr 16 2012

[n: n in [1..1500] | forall{d: d in PrimeDivisors(n) | d mod 5 eq 1}]; // Vincenzo Librandi, Aug 21 2012

ok[1]=True;ok[n_]:=And@@(Mod[#,5]==1&)/@FactorInteger[n][[All,1]];Select[Range[2000],ok] (* Vincenzo Librandi, Aug 21 2012 *)
Select[Range[1000],Union[Mod[#,5]&/@FactorInteger[#][[All,1]]]=={1}&] (* Harvey P. Dale, Apr 19 2019 *)

is(n)=#select(p->p%5!=1, factor(n)[,1])==0 \\ Charles R Greathouse IV, Mar 11 2014

A206291 merged in by Franklin T. Adams-Watters, Sep 21 2012

A216891 Decimal expansion of the minimal zero x(1) of the function F(x) = f(f(x)) - x, where f(x) = cos(x) - sin(x).

Original entry on oeis.org

8, 3, 0, 1, 9, 8, 5, 1, 7, 0, 6, 7, 8, 2, 3, 9, 3, 4, 5, 5, 2, 2, 5, 6, 2, 7, 1, 9, 5, 5, 2, 7, 1, 0, 6, 5, 7, 7, 8, 2, 0, 6, 3, 0, 8, 4, 3, 9, 4, 5, 4, 3, 7, 3, 1, 9, 3, 9, 5, 5, 2, 4, 1, 2, 2, 1, 6, 0, 8, 4, 8, 3, 2, 0, 4, 5, 6, 1, 8, 8, 9, 6, 2, 2, 6, 4, 1, 6, 3, 8, 6, 9, 7, 2, 6, 2, 9, 1, 2, 1, 5, 9, 1, 2, 3

Offset: 0

Roman Witula, Sep 19 2012

Let x(2) and x(3) denote the remaining zeros of F(x), x(2) < x(3). Then it could be proved that f(x(1)) = x(3), f(x(3)) = x(1), and f(x(2)) = x(2).
The decimal expansions of x(2) and x(3) in A206291 and A216863 respectively are presented.
We note that the plot of the restriction of F(x) to the interval [-2,2] "is very similar" to the plot of the polynomial (x-x(1))*(x-x(2))*(x-x(3)) for x in [-2,2].
Let A = {x in R: f^n(x) = x(2) for some nonnegative integer n, where f^n denotes the n-th iteration of f}. Then if z is a real number, which does not belong to A, and z(0):= z, z(n+1) = f(z(n)) = sqrt(2)*sin(Pi/4 - z(n)), n in N, then one of the subsequences either {z(2*n-1)} or {z(2*n)} is convergent to x(1) and the second one is convergent to x(3).

			We have x(1) = -0.830198517...

R. Witula, D. Slota and Szeged Problem Group "Fejentalaltuka", An Iteration Convergence: 11318[2007,745], Amer. Math. Monthly, 116 No 7 (2009), 648-649.

Cf. A216863, A206291, A215668, A215670, A215832, A215833, A168546.

A216863 The decimal expansion of the maximal zero x(3) of the function F(x) = f(f(x)) - x, where f(x) = cos(x) - sin(x).

Original entry on oeis.org

1, 4, 1, 2, 7, 9, 4, 5, 8, 5, 7, 2, 7, 6, 2, 2, 2, 4, 5, 2, 9, 3, 5, 7, 7, 5, 8, 6, 5, 0, 6, 6, 3, 7, 6, 3, 2, 1, 2, 4, 5, 5, 8, 8, 9, 1, 2, 7, 3, 8, 1, 5, 1, 6, 5, 8, 6, 4, 1, 7, 7, 5, 2, 5, 5, 3, 0, 1, 1, 2, 5, 2, 4, 7, 7, 1, 2, 2, 0, 5, 5, 1, 8, 2, 4, 3, 3, 4, 0, 8, 7, 8, 5, 0, 0, 6, 0, 8, 9, 2, 5, 6, 1, 5, 0

Offset: 1

Roman Witula, Sep 18 2012

The decimal expansions of the only other zeros x(1) and x(2) of F(x) are given in A216891 and A206291. See the comments in A216891 for more information about intriguing properties of these zeros.

			We have x(3) = 1.4127945857276222... < sqrt(2).

R. Witula, D. Slota and Szeged Problem Group "Fejentalaltuka", An Iteration Convergence: 11318[2007, 745], Amer. Math. Monthly, 116 No 7 (2009), 648-649.

Cf. A216891, A206291, A215668, A215670, A215832, A215833, A168546.

f[x_] := Cos[x] - Sin[x];
FindRoot[f[f[x]] - x, {x, 1.4}, WorkingPrecision -> 110] (* T. D. Noe, Sep 24 2012; first line by Rick L. Shepherd, Jan 04 2014 *)

default(realprecision,110);
f(x) = cos(x) - sin(x);
solve(x = 1.4, 1.5, f(f(x)) - x) \\ Rick L. Shepherd, Jan 03 2014

Offset corrected by Rick L. Shepherd, Jan 03 2014

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A004615 Divisible only by primes congruent to 1 mod 5.

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A216891 Decimal expansion of the minimal zero x(1) of the function F(x) = f(f(x)) - x, where f(x) = cos(x) - sin(x).

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A216863 The decimal expansion of the maximal zero x(3) of the function F(x) = f(f(x)) - x, where f(x) = cos(x) - sin(x).

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